#453912

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16216
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,8,16,29)(4,9)(5,10)(6,25)(7,11)(12,13)(14,40,42,27)(15,43,22,28)(17,37,26,30)(18,31,46,57)(19,32)(20,50)(21,33)(23,53)(24,35)(34,38)(36,39)(41,54,51,62)(44,55,52,60)(45,63,47,56)(48,64)(49,58)(59,61);(1,3,13,17,7,16,6,26)(2,8,25,30,11,29,12,37)(4,18,38,42,21,46,20,14)(5,22,23,44,24,15,39,52)(9,31,50,40,33,57,34,27)(10,28,36,60,35,43,53,55)(19,47,48,51,49,45,61,41)(32,56,59,54,58,63,64,62);(1,4,6,20,7,21,13,38)(2,9,25,50,11,33,12,34)(3,14,17,18,16,42,26,46)(5,19,23,48,24,49,39,61)(8,27,37,57,29,40,30,31)(10,32,53,64,35,58,36,59)(15,41,44,45,22,51,52,47)(28,54,60,63,43,62,55,56);(1,5)(2,10,11,35)(3,15)(4,19)(6,23)(7,24)(8,28,29,43)(9,32,33,58)(12,36,25,53)(13,39)(14,41)(16,22)(17,44)(18,45)(20,48)(21,49)(26,52)(27,54,40,62)(30,55,37,60)(31,56,57,63)(34,59,50,64)(38,61)(42,51)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=efg,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361108);